The Structure of a Lie Algebra Attached to a Unit Form
Received:November 04, 2017  Revised:September 01, 2018
Key Word: Nakayama algebras   finite dimensional simple Lie algebras   Ringel-Hall Lie algebras
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11571360) and the Natural Science Foundation of Fujian Province (Grant Nos.2016J01006; JZ160427).
 Author Name Affiliation Yalong YU College of Mathematics and Informatics, Fujian Normal University, Fujian 350117, P. R. China Zhengxin CHEN College of Mathematics and Informatics, Fujian Normal University, Fujian 350117, P. R. China
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Let $n\geq 4$. The complex Lie algebra, which is attached to the unit form $\mathfrak{q}(x_1,x_2,\ldots, x_n)=\sum_{i=1}^nx_i^2-(\sum_{i=1}^{n-1}x_ix_{i+1})+x_1x_n$ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type $\mathbb{D}_n$, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.